a) $$ 2 a-5 b+3 a+8 b $$ b) $$ a-5 a+9 b+6 a-7 b $$ c) $$ 2 x-y-x+5 x-6 y $$ d) $$ 2 a+3 b-(a+5 c)+(2 b-4 b)+3 c+8 a $$

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**a) $$2a - 5b + 3a + 8b$$**

1. Group like terms:
   - $$a$$-terms: $$2a + 3a = 5a$$
   - $$b$$-terms: $$-5b + 8b = 3b$$

2. Simplified expression:  
   $$5a + 3b$$

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**b) $$a - 5a + 9b + 6a - 7b$$**

1. Group like terms:
   - $$a$$-terms: $$a - 5a + 6a = (1 - 5 + 6)a = 2a$$
   - $$b$$-terms: $$9b - 7b = 2b$$

2. Simplified expression:  
   $$2a + 2b$$

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**c) $$2x - y - x + 5x - 6y$$**

1. Group like terms:
   - $$x$$-terms: $$2x - x + 5x = (2 - 1 + 5)x = 6x$$
   - $$y$$-terms: $$-y - 6y = -7y$$

2. Simplified expression:  
   $$6x - 7y$$

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**d) $$2a + 3b - (a + 5c) + (2b - 4b) + 3c + 8a$$**

1. Remove the parentheses, taking care with the negative sign:
   $$
   2a + 3b - a - 5c + 2b - 4b + 3c + 8a
   $$

2. Group like terms:
   - $$a$$-terms: $$2a - a + 8a = (2 - 1 + 8)a = 9a$$
   - $$b$$-terms: $$3b + 2b - 4b = (3 + 2 - 4)b = 1b$$  
     (which is simply $$b$$)
   - $$c$$-terms: $$-5c + 3c = -2c$$

3. Simplified expression:  
   $$9a + b - 2c$$
a) $$5a + 3b$$ b) $$2a + 2b$$ c) $$6x - 7y$$ d) $$9a + b - 2c$$

a) \( 2 a-5 b+3 a+8 b \) b) \( a-5 a+9 b+6 a-7 b \) c) \( 2 x-y-x+5 x-6 y \) d) \( 2 a+3 b-(a+5 c)+(2 b-4 b)+3 c+8 a \)

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